The city parks agency yesterday began soliciting bids for sponsor rights to 631 basketball courts and 55 dog runs in hopes of pulling in at least $5 million a year for its operations.

Producers of sugar-laden soft drinks like Coca-Cola aren’t prohibited from bidding. But they’re not being encouraged, either.

“Companies should be aware the city would be unlikely to do something in conflict with the public interest,” said Julie Wood, a spokeswoman for Mayor Bloomberg.

Hint, hint.

Sources said Coke wouldn’t have a shot of having its name and logo adorn city property. Diet Coke? Well, that remains a question mark, since it isn’t included in the mayor’s proposed ban of sugary beverages larger than 16 ounces.

The Parks Department first floated the idea of peddling naming rights for some of its facilities in 2008, but withdrew the scheme when the economy tumbled.

The latest idea is to offer the dog runs for at least $1.5 million and the basketball courts for $3.5 million.

Like many other city agencies, the Parks Department is facing proposed budget cuts — in its case, $33.4 million.

As the mayor was stepping up his campaign against waistline-expanding sodas, he encountered one skeptic right within his own family.

Bloomberg’s independent-minded younger daughter, Georgina, 29, told a New York magazine reporter at a social event Thursday night that the best method for promoting healthier eating is to make healthy foods less expensive.

“His intentions are good,” she said of her father’s anti-soda crusade. “And it’s the kind of thing the public will thank him for in 20 years. However, people should be allowed to make their own bad choices.”

Georgina, a champion equestrian athlete, suggested that, ideally, her father would “lower the cost of healthy food instead of banning unhealthy food.”